In a digital communication system the transmitter takes information bits to be communicated and converts them into a form suitable for transmission to the receiver. This is performed by encoding (protecting) information bits at the transmitter with an error correcting code and subsequently mapping the coded bits to a sequence of symbols. The symbols are then transmitted over a communication channel to a receiver. The receiver captures the channel output and tries to recover the information bits. Hence, a practical transmission scheme is described by an error-correcting code and means for performing a mapping function.
The channel considered in the context of the present invention includes fading experienced between the transmitter and the receiver and Gaussian distributed noise added to the signal at the receiver. In many applications, such as frequency-hopping schemes (GSM, EDGE), the channel model is a block fading (BF) channel, where for example due to a delay-constraint, a transmitted packet is affected by a limited number of channel states, each subject to independent flat fading. The BF channel can be represented by B fading gains, where B is an integer number expressing the ratio between a code word duration and the duration over which the fading remains constant. Other applications include cooperative communications, multiple input multiple output (MIMO) slow fading channels and frequency-selective channels combined with OFDM, . . . .
In communications the application often requires a transmission at a certain spectral efficiency with a certain quality of transmission. These requirements can be achieved by any transmission scheme, but at a certain price, i.e., the transmitted energy per information bit. It is the aim of any communication engineer to minimize the required energy for a certain quality of transmission and a certain spectral efficiency. One measure for the quality of transmission and hence for the performance of the applied communication scheme is the word error rate (WER), which is the fraction of the total number of transmitted packets (i.e. code words) which has not been decoded correctly at the receiver. An important factor of the WER is the diversity order d, because the WER is inversely proportional to the signal-to-noise ratio (SNR) to the power d. When the maximum diversity order is obtained, one says that the coded system achieves full diversity. The maximum diversity order obtainable over a slow fading channel with B fading gains is B. The average word error rate performance of a communication system has an information-theoretical lower bound determined by the outage probability limit. In order to approach this lower bound, full-diversity should be achieved. Further, the horizontal SNR gap between the outage probability and WER must be decreased as much as possible.
Capacity-achieving codes, such as turbo codes and low-density parity-check (LDPC) codes are well known in the art, see e.g. the handbook Modern Coding Theory by T. J. Richardson and R. L. Urbanke, Cambridge University Press, 2008. Many excellent capacity-achieving codes are known for standard channels, such as the additive white Gaussian noise (AWGN) channel. However, few capacity-achieving codes are known for the BF channel. In order to approach the outage probability limit as closely as possible, it is needed to optimize the parameters of capacity-achieving codes.
Binary LDPC codes are powerful capacity-achieving linear error-correcting codes. Linear error-correcting codes are completely determined by their parity-check matrix H of dimension N−K×N. Random LDPC codes have at least one sparse low-density parity-check matrix and the number of ones in a row, resp. column, follow a certain distribution. The fraction of ones in the matrix, being in a column with i ones, is given by λi. The fraction of ones in the matrix, being in a row with i ones, is given by ρi. Hence, LDPC codes are parameterized by two polynomials
                    λ        ⁡                  (          x          )                    =                        ∑                      i            =            2                                d            b                          ⁢                                  ⁢                              λ            i                    ⁢                      x                          i              -              1                                            ;              ρ      ⁡              (        x        )              =                  ∑                  i          =          2                          d          c                    ⁢                          ⁢                        ρ          i                ⁢                  x                      i            -            1                                                  where        ⁢                                  ⁢                              ∑                          i              =              2                                      d              b                                ⁢                                          ⁢                      λ            i                              =                                    ∑                          i              =              2                                      d              c                                ⁢                      ρ            i                          =        1              ,  and where db and dc are the maximum number of ones per column and per row, respectively. LDPC codes satisfying the degree distributions λ(x) and ρ(x) are said to belong to the same ensemble of LDPC codes. Note that many different parity-check matrices, denoted as instances, can verify the same distributions λ(x) and ρ(x). However, when the block length N goes to infinity, the WER performance of all these different instances converges exponentially fast to a value which only depends on the degree distributions. For a specific code instance, the parity-check matrix can be graphically represented by a bipartite graph, denoted as a Tanner graph, displaying bit nodes or variable nodes at the left side and check nodes at the right side. Each bit node has db edges connected to it and each check node has dc edges connected to it. In practice, this graph has cycles and is not a tree.
A constraint valid for all full-diversity codes is the coding rate Rc, which is limited by the Singleton bound (e.g. Rc≦0.5 when B=2). Next, depending on the error-correcting code type, additional requirements may be needed to obtain full-diversity. For example, root LDPC codes (as presented in the paper “Low-Density Parity-Check Codes for Nonergodic Block-Fading Channels” (J. J. Boutros et al., IEEE Trans. Information Theory, vol. 56, no. 9, pp. 4286-4300, September 2010)), special semi-random LDPC codes, are full-diversity LDPC codes.
The above-mentioned Boutros paper introduced a new family of LDPC codes, called root LDPC codes, that achieve full-diversity on block fading (BF) channels. However, these root LDPC codes are different from standard random LDPC codes and therefore, the traditional tools for coding gain optimization, such as standard EXIT charts, cannot be directly used for root LDPC codes. In another paper by Boutros (“Diversity and coding gain evolution in graph codes”, Information Theory and Applications (ITA), San Diego, 2009) it was proved that full-diversity standard random LDPC codes do not exist for the maximum coding rate, given by the Singleton bound.
However, full-diversity standard random LDPC codes may exist for coding rates close to the maximum coding rate. Furthermore, near full-diversity standard random LDPC codes for maximum rate may also exist. Codes achieving near full-diversity are defined as codes whose WER is inversely proportional to the SNR to the power d, where d equals the maximum diversity order and as long as the error rate is larger than a certain error rate of interest. For example, codes may achieve this maximum decrease in error rate with SNR for word error rates greater than 10−4, which may be sufficient for most applications. Consequently, there is a need for a method for optimizing the code parameters of standard random LDPC codes in the context of BF channels, so that the WER is minimized.
The optimization technique here above can yield codes that approach the outage probability very closely, for a given modulation. However, using multidimensional modulations with linear precoding, the outage probability associated with a discrete input alphabet can be minimized. Furthermore, the Singleton bound is modified when using precoding, so that full-diversity can be achieved for larger coding rates. The design of multidimensional modulations has been extensively studied for uncoded transmission schemes. In some recent work coded transmission schemes have been studied.
The paper “Rotated Modulations for Outage Probability Minimization: a fading space approach” (D. Duyck et al., Intl Symposium on Information Theory, Austin, Tex., USA, June 2010, pp. 1061-1065) deals with the effect of linear precoding on the outage probability. In the paper the size M and the multidimensional shape of the constellation are determined so that the outage probability of a block fading channel with a discrete input approaches the outage probability of a block fading channel with continuous Gaussian distributed input very closely. A continuous Gaussian distributed input yields the smallest outage probability. Hence, as stated in the conclusions of the paper, a close to optimum theoretical lower bound is provided. The labelling of the multidimensional constellation and the design of error-correcting codes with linear precoding are still to be elaborated.
Consequently, there is a need for a method for finding an improved LDPC coded modulation scheme with given spectral efficiency for communication over block fading channels. Such method may serve in cases with or without precoding, and may also be a basis for the design of practical coded systems in other scenarios, such as MIMO channels and cooperative communications. Further, there is a need for a transmitter device with a coded modulation scheme having parameters so configured that the coding gain of practical schemes is improved.
Optimizations of coded modulations with LDPC codes for Gaussian channels are known in the art. For Gaussian channels, whereby the fading gain is not random and remains constant all the time, codes and mapping exist so that the theoretical lower bound (i.e. the channel capacity) is approached closely. The paper “Design of Low-Density Parity-Check Codes for Modulation and Detection” (ten Brink et al., IEEE Trans. Comm., vol. 52, no. 4, pp. 670-678, April 2004) studies a coding and modulation technique where the coded bits of an irregular low-density parity check code are passed directly to a mapper. The code is optimized by performing a curve fitting on extrinsic information transfer (EXIT) charts. Design examples are given for an additive white Gaussian noise (AWGN) channel. The authors mention it is not known how to optimize these coded modulations for fading channels, in other words, how to take into account the randomness of the fading gain. Also, it was stated in the Ph.D. thesis of A. Guillén i Fàbregas (“Concatenated codes for block-fading Channels”, EPFL, June 2004), that EXIT charts are not a suitable tool for studying error performance on BF channels. This is the reason why the authors in the Boutros paper “Low-Density Parity-Check Codes for Nonergodic Block-Fading Channels” did not know how to optimize the degree distributions of LDPC codes for BF channels, and therefore simply simulated the performance of many state of the art codes, to conclude they don't perform well on the BF channel. Next, the authors introduced a new family of LDPC codes, achieving full-diversity, but the degree distributions of this family were never optimized either.
As no solution for standard random LDPC codes on slow fading channels is available, codes and mappings well performing for the Gaussian channel have sometimes been used on the slow fading channel. The performance is poor, because the solution is not adapted to the context. Also root LDPC codes are used on the BF channel. However, they have not been extended and simulated for other coding rates than Rc=0.5. Furthermore, linear-time encoding with root LDPC codes is not trivial and standard optimization techniques, such as EXIT charts, have not been modified yet for root LDPC codes. Finally, off the shelf standard random LDPC codes cannot be used, which increases the implementation cost of root LDPC codes for chip designers.
The authors of the above-mentioned ten Brink paper on LDPC codes are also the inventors of the digital transmission system and method disclosed in U.S. Pat. No. 6,662,337. The invention describes the turbo principle in iterative joint demapping and decoding with soft values where the output of the demapper is an input for the decoder and vice versa. It also proposes two mappings that are mixed adaptively dependent on the channel conditions and the number of iterations to be used. The applied criterion for best mapping is performed for Gaussian channels.